The comments are all good - and in answer to your question "Will glueing a non-stretchy piece on stretchy fabric cause tearing?" the answer is most likely yes, or at a minimum, it can cause "runs" (torn yarns that pull through the bodysuit fabric). Also, when the bodysuit fabric stretches, the glued decal might just pop off.
Your potential decal materials from foam, iron-on, and faux-leather will behave quite differently from each other. The foam will have some stretch, the iron-on and faux-leather will have no to very insignificant stretch. You may need different attachment mechanisms for each type of patch.
Even if your decals are stretchy (e.g., thin foam), they may not have have the same stretch factor and in the same direction as the fabric of your bodysuit.
One thing that might work is to adhere the decals with spots of glue or a few tacking stitches (one stitch tied off), sort of like spot welding. This will still allow some stretch of the suit fabric underneath. The fewer attachment points the better to allow for base-fabric stretch, but the more attachments points you have will probably look better.
Another thing to try would be to loosely attach the outline of the patch with running stitches using clear elastic thread. Use as few stitches as possible to still achieve the look you want.
Clear Elastic Thread
Good luck! Let us know what you try and how it works out!